i've broken three or four ties between POTM and The Sm4him Challenge. One method I use is to hold my vote until the end, and break the tie that way. I have also consulted others. Dean will do what works best for him.
We had a tie (three-way IIRC) on Photo of the Year. For that, we held a second round of voting with just the tied entries.
Change You Vote is fine with me. Some folks may see something in one or another after they have voted, and I can see where someone might hit the wrong box, especially on a phone. I have a small screen and rather large (fat) fingers and frequently hit the wrong marker for things.
I stopped the "Display Votes Publicly" early on when I realized it showed who voted for what.